D.L. Kreher. Department of Mathematical Sciences. Michigan Technological University. and. D.R. Stinson

Transcription

1 Pseucode: A L A T E X Style File for Displaying Algorithms D.L. Kreher Department of Mathematical Sciences Michigan Technological University Houghton, MI kreher@mtu.edu and D.R. Stinson Department of Combinatorics and Optimization University of Waterloo Waterloo ON, N2L 3G1 dstinson@uwaterloo.ca September 3, Introduction This paper describes a LATEX environment named pseucode that can be used for describing algorithms in pseucode form. This is the style used in our textbook Combinatorial Algorithms: Generation, Enumeration and Search [2]. The style le pseucode.sty is available for free wnloading from the web page This package is quite easy to use, and allows algorithms to be described in a LATEX cument using a natural Pascal-like syntax. In the remaining sections of this note, we describe how to use the pseucode environment and we present some examples. Readers familiar with LATEX (see [3]) should be able to easily customize the style le to include additional desired features. The pseucode environment requires the fancybox package by Timothy Van Zandt. This package is described in Section of [1]. Other environments for describing algorithms include alg, algorithmic, newalg 1

2 and program. These style les, as well fancybox, are all available from the CTAN web site 2 The pseucode Environment Within the pseucode environment, a number of commands for popular algorithmic constructs are available. In general, the commands provided can be nested to describe quite complex algorithms. The pseucode environment is invoked as follows: \begin{pseucode}{<name>}{<parameters>} pseucode constructs \end{pseucode} The argument <Name> is the name of the algorithm, and <Parameters> is a list of parameters for the algorithm. For example, the commands \begin{pseucode}{celsiustofahrenheit}{c} f \GETS {9c/5} + 32\\ \RETURN{f} \end{pseucode} produce the following output when included in a LATEX cument: Algorithm 2.1: CelsiusToFahrenheit(c) f 9c= return (f) Notice that the command \GETS produces a left arrow, which we use to indicate an assignment of a variable. The user could use instead some other symbol, if desired. For example, \GETS could be replaced by =, as is ne in the \C" programming language. 2.1 The begin-end Construct To form compound statements from simple statements, the begin-end construct is used as follows: some statement\\ another statement\\ yet another statement 2

3 This generates the following: 8 < : some statement another statement yet another statement The eect of this construct is to group a collection of statements using a left brace bracket of the appropriate size. In the sections that follow we will use the notation <stmt> to indicate a simple statement or a compound statement. Note that the contents of statements are typeset in math mode. Therefore, non-math mode text must be enclosed in an \mbox{}. Observe that the uble backslash \\ plays the same role as the semicolon in Pascal, i.e., it is used to separate statements, and should never appear before. 2.2 The if-then-else Construct The if-then-else construct takes various forms, such as the following: \IF <condition> \THEN <stmt> \IF <condition> \THEN <stmt> \ELSE <stmt> \IF <condition> \THEN <stmt> \ELSEIF <stmt> \THEN <stmt> Note that there is no limit placed on the number of \ELSEIFs that may be used in an if-then-else construct. For example, the commands: \IF some condition is true \THEN some statement\\ another statement\\ yet another statement \ELSEIF some other condition is true \THEN some statement\\ another statement\\ yet another statement \ELSEIF some even more bizarre condition is met \THEN something else \ELSE the default actions 3

4 would produce the following output: 8 if some condition is true < some statement then another statement : yet another statement else if8 some other condition is true < some statement then another statement : yet another statement else if some even more bizarre condition is met then something else else the default actions 2.3 The for Loop The for loop takes the following forms: \FOR <var> \GETS <lower> \FOR <var> \GETS <upper> \FOREACH <condition> \DO For example, \TO <upper> \DO <stmt> \DOWNTO <lower> \DO <stmt> <stmt> \FOR i \GETS 0 \TO 10 \DO some processing produces for i 0 to 10 some processing and \FOREACH x \in \mathcal{s} some processing \DO produces for each x 2 S some processing 2.4 The while Loop The while loop takes the following form: \WHILE <condition> \DO <stmt> For example, 4

5 \WHILE some condition holds \DO some processing produces while some condition holds some processing 2.5 The repeat-until Loop The repeat-until loop takes the following form: \REPEAT <stmt> \UNTIL <condition> For example, \REPEAT some processing \UNTIL some condition is met produces repeat some processing until some condition is met 2.6 Main Programs and Procedures We can describe a main program that calls one (or more) procedures as follows: \begin{pseucode}{<name>}{<parameters>} \PROCEDURE{<ProcedureName>}{<ProcedureParameters>} some stuff PROCEDURE \MAIN some stuff\\ \CALL{<ProcedureName>}{<ActualParameters}>\\ more stuff MAIN \end{pseucode} Here is a simple example to illustrate the use of a main program calling a procedure. The commands 5

6 \begin{pseucode}{temperaturetable}{lower, upper} \PROCEDURE{CelsiusToFahrenheit}{c} f \GETS {9c/5} + 32\\ \RETURN{f} PROCEDURE \MAIN x \GETS lower \\ \WHILE x \leq upper \DO \OUTPUT{x, \CALL{CelsiusToFahrenheit}{x}}\\ x \GETS x+1 MAIN \end{pseucode} produce the following output: Algorithm 2.8: TemperatureTable(lower; upper) procedure CelsiusToFahrenheit(c) f 9c= return (f) main x lower while x upper output (x; CelsiusToFahrenheit(x)) x x Comments A comment statement may be inserted in an algorithm using the following command: \COMMENT{<stmt>} For example, the commands A \GETS B\\ \COMMENT{Now increment the value of $A$}\\ A \GETS A+1 produce the output 6

7 A B comment: Now increment the value of A A A + 1 Note that comments are assumed to be text. Thus, in order to include mathematical expressions in a comment, math mode must be used explicitly. 2.8 Other Predened Keywords Several other predened keywords are available. We summarize their usage in Table 1. Table 1: Other Predened Keywords command \LOCAL{list of variables} \GLOBAL{list of variables} \EXTERNAL{list of procedures} \RETURN{list of values} \OUTPUT{list of values} \EXIT \AND \OR \NOT \TRUE \FALSE \GETS output local list of variables global list of variables external list of procedures return (list of values) output (list of values) exit and or not true false Also note that all of the keywords \IF, \WHILE, \CALL{}{}, \NOT, etc. are available for use outside of the pseucode environment, but they must be input in math mode. For example, generates The $\WHILE$ loop is our friend. The while loop is our friend. 2.9 Statement Numbering Statements can be numbered and given a reference key so that they can be referenced in a LATEX cument using a \ref{} command (see section 4.2 of [3]). This is ne as follows: 7

8 \STMTNUM{<space>}{<key>} The argument <space> is the amount of space to be left between the text and the statement number. This is a length that is specied by the user, and generally will require some experimentation in order for it to look nice. The argument <key> is the reference key used in the LaTeX \ref{} command to refer to the given statement. The default numbering for statements is arabic. However, it can be changed by a suitable \renewcommand{}. An example is provided in the next section. 3 An example The following example demonstrates the use of the pseucode environment to describe a complete algorithm, the familiar \mergesort" algorithm. The LATEX input \renewcommand{\thepseunum}{\roman{pseunum}} \begin{pseucode}{mergesort}{n,x} \label{mergesort} \COMMENT{Sort the array $X$ of length $n$}\\ \IF n=2 \THEN \IF X[0]>X[1] \THEN T \GETS X[0]\\ X[0]\GETS X[1]\\ X[1]\GETS T \ELSEIF n>2 \THEN m\gets \lfloor n/2 \rfloor\\ \FOR i\gets 0 \TO m-1 \DO A[i] \GETS X[i]\\ \FOR i\gets m \TO n-1 \DO B[i] \GETS X[i]\\ \COMMENT{Now sort the subarrays $A$ and $B$}\\ \CALL{MergeSort}{m,A}\\ \CALL{MergeSort}{n-m,B}\\ i\gets 0\\ j\gets 0\\ \FOR k \GETS 0 \TO n-1 \DO \IF A[i] \leq B[j] \THEN 8

9 X[k]\GETS A[i] \STMTNUM{1in}{st.1}\\ i\gets i+1 \ELSE X[k]\GETS B[j] \STMTNUM{1.03in}{st.2}\\ j\gets j+1 \end{pseucode} produces the following output: Algorithm 3.1: MergeSort(n; X) comment: Sort the array X of length n 8 if n = 2 if X[0] > 8X[1] < T X[0] then then X[0] X[1] : X[1] T else if n > 2 8 m bn=2c for i 0 to m? 1 A[i] X[i] for i m to n? 1 B[i] X[i] comment: Now sort the subarrays A and B then MergeSort(m; A) MergeSort(n? m; B) i 0 j 0 for k 8 0 to n? 1 if A[i] B[j] X[k] A[i] (i) then i i + 1 X[k] B[j] (ii) else j j + 1 The counter pseunum keeps track of the statement numbers. The style of the counter values can be changed using the method described in Section 6.3 of [3]. For example, we used the command 9

10 \renewcommand{\thepseunum}{\roman{pseunum}} in our example so that statements were numbered with lowercase Roman numerals. We also assigned a label to the algorithm using the \label{} command that is described in Section 4.2 of [3]. Finally, by trial and error, we determined spacing so that the statement numbers would be vertically aligned. We now give an example of how the numbered statements in the above algorithm can be referenced in a LATEXcument. The commands On lines (\ref{st.1}) and (\ref{st.2}) of Algorithm \ref{mergesort}, we determine the $k$th element of the sorted array. produce the following output: On lines (i) and (ii) of Algorithm 3.1, we determine the kth element of the sorted array. 4 Framing The pseucode environment also has an optional parameter, <frame>. The complete form of the pseucode environment is \begin{pseucode}[<frame>]{<name>}{<parameters>} pseucode constructs \end{pseucode} The possible values of <frame> are: shawbox ublebox ovalbox Ovalbox framebox plain ruled display The values ending with \box" draw various types of frames around the algorithm. The value plain is the default and adds no frame to the algorithm. The value display is used for displaying sections of code without the algorithm name or parameters. Here are some examples with input: \begin{pseucode}[<frame>]{squareandmultiply}{x,b,n} \COMMENT{ Compute $x^b \pmod{n}$}\\ z\gets 1\\ \WHILE b > 0 \DO z \GETS z^2 \pmod{n} \\ \IF b\mbox{ is odd} 10

11 \THEN z \GETS z \ct x \pmod{n} \\ b \GETS \CALL{ShiftRight}{b} \\ \RETURN{z} \end{pseucode} where we give <frame> each of the values described above. When <frame> is shawbox we obtain: Algorithm 4.1: SquareAndMultiply(x; b; n) comment: Compute x b (mod n) z 1 while8 b > 0 z z 2 (mod n) return (z) When <frame> is ublebox we obtain: Algorithm 4.2: SquareAndMultiply(x; b; n) comment: Compute x b (mod n) z 1 while8 b > 0 z z 2 (mod n) return (z) When <frame> is ovalbox we obtain: 11

12 Algorithm 4.3: SquareAndMultiply(x; b; n) comment: Compute x b (mod n) z 1 while8 b > 0 z z 2 (mod n) return (z) When <frame> is Ovalbox we obtain: Algorithm 4.4: SquareAndMultiply(x; b; n) comment: Compute x b (mod n) z 1 while8 b > 0 z z 2 (mod n) return (z) When <frame> is framebox we obtain: Algorithm 4.5: SquareAndMultiply(x; b; n) comment: Compute x b (mod n) z 1 while8 b > 0 z z 2 (mod n) return (z) When <frame> is plain or if [<frame>] is omitted we obtain: 12

13 Algorithm 4.6: SquareAndMultiply(x; b; n) comment: Compute x b (mod n) z 1 while8 b > 0 z z 2 (mod n) return (z) When <frame> is ruled we obtain: Algorithm 4.7: SquareAndMultiply(x; b; n) comment: Compute x b (mod n) z 1 while8 b > 0 z z 2 (mod n) return (z) The purpose of the value display is to allow portions of algorithms to be displayed with out the algorithm header. Thus for example to display the section of code in the while loop of the SquareAndMultiply() algorithm one could write \begin{center} \begin{minipage}{2in} \begin{pseucode}[display]{}{} z \GETS z^2 \pmod{n} \\ \IF b\mbox{ is odd} \THEN z \GETS z \ct x \pmod{n}\\ b \GETS \CALL{ShiftRight}{b} \end{pseucode} \end{minipage} \end{center} which would produce the following output: 13

14 z z 2 (mod n) References [1] M. Goossens, F. Mittelbach and A. Samarin, The LA TE XCompanion, Addison-Wesley, [2] D.L. Kreher and D.R. Stinson, Combinatorial Algorithms: Generation, Enumeration and Search, CRC Press, [3] L. Lamport, LATEX, A Document Preparation System, Addison-Wesley,